Nearest polynomial with given properties has many applications in control theory and applied mathematics. Given a complex univariate polynomial f（z） and a zero α, in this paper we explore the problem of computing a...Nearest polynomial with given properties has many applications in control theory and applied mathematics. Given a complex univariate polynomial f（z） and a zero α, in this paper we explore the problem of computing a complex polynomial f^^（z） such that f^^（a）=0 and the distance ｜｜f^^-f｜｜ is minimal. Considering most of the existing works focus on either certain polynonfial basis or certain vector norm, we propose a common computation framework based on both general polynomial basis and general vector norm, and summarize the computing process into a four-step algorithm. Further, to find the explicit expression of f（z）, we focus on two specific norms which generalize the familiar lp-norm and mixed norm studied in the existing works, and then compute f^^（z） explicitly based on the proposed algorithm. We finally give a numerical example to show the effectiveness of our method.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.6143200311171052+3 种基金612723716132820611361005)the Research Programs of Gannan Normal University(Grant No.14zb21) College of Mathematics and Computer Science
文摘Nearest polynomial with given properties has many applications in control theory and applied mathematics. Given a complex univariate polynomial f（z） and a zero α, in this paper we explore the problem of computing a complex polynomial f^^（z） such that f^^（a）=0 and the distance ｜｜f^^-f｜｜ is minimal. Considering most of the existing works focus on either certain polynonfial basis or certain vector norm, we propose a common computation framework based on both general polynomial basis and general vector norm, and summarize the computing process into a four-step algorithm. Further, to find the explicit expression of f（z）, we focus on two specific norms which generalize the familiar lp-norm and mixed norm studied in the existing works, and then compute f^^（z） explicitly based on the proposed algorithm. We finally give a numerical example to show the effectiveness of our method.
